DERIVATION OF CONSTANTS OF A n-UNIT CASCADE TRANSFORMER

BY B. N. JAYARAM

(High Voltage Engineering Laboratory, Department of Power Engineering, Indian Institute of Science, Bangalore-3)

Received January 27, 1955

SUMMARY

Expressions for the open-circuit admittance and short-circuit impe- dance of a n-unit cascade transformer have been derived. These enable the estimation of the above constants from a knowledge of the constants of any one transformer and the arrangement of the transformers.

1. INTRODUCTION

It is the standard practice in High Voltage Laboratories' to produce high voltage a.c. at power frequencies, above 500 kV, by Cascade Transformers." Cascading of transformers provides a connection whereby the h.t. windings of several similar transformers are virtually placed in series whilst only the primary 'est: of the first (ground end) transformer is connected to the supply, the power for the higher members of the group being provided by the first and intermediate trans- formers through a third winding known as the exciting winding. This winding has q1:1 ratio with the primary, i.e., low voltage winding, but is insulated from the tank upto the full voltage of the secondary, i.e., high voltage winding. The con- nection diagram of a 3-unit cascade transformer to provide 1050 kV, which has been installed at the High Voltage Laboratory,' Indian Institute of Science, Banga- lore, is shown in Fig. I.

Since each unit of a cascade transformer is a three-winding transformer, for purposes of calculation of regulation, etc., it can be represented by an equivalent circuit'' , 5 shown in Fig. 2, in which the three impedances ZVI, ZE and ZH represent the leakage impedances associated with the three windings, low voltage (00, exciting (E) and high voltage (11). Yx represents the excitation admittance of each transformer. Recently mention has been made by Hagenguth and others' that the ratio of transformers connected in cascade is calculable on the basis of a series connection of three-winding transformer equivalent circuits. In the present paper, with the idea of generalisation, the theory has been extended to a n-unit cascade transformer, in which 'ft' identical transformers are connected in the Dessauer cascade connectionst 3 Expressions fot the open-circuit admittance and short-circuit impedance have been derived. The results of testi conducted on the cascade transformers at the Indian Institute of Science will be published later. _ Ny. :t. • 1/4

I ',se t

162. _ . . - • -

Constants of a n-Unit Cascade Transformer 163

2.3 kV

GNU

t ri I 1 1 i 1 i cD I 1

I I I IV

- I H ym 1 I TAP - FitH .

on- CUT st. TAri k or

2-3 icV

iI'i t..v . -

>k* .

.

k

La TAP on

TAN K or • r

1 a 1 I

I - 1 -.. 1

-- I • I

i I

II

2-3 kV

r

> x 0

R

r

: r

® I II

r I

. _ .

LII . • . ' ..

- v.H. : 1"Ap

rm. it opt m)-3 Il l

fIG.1 . C•AwecnON Ottani or Mr 3 -usir C4scaDif nfitterOANER rivear4141P 47 TAT 2'ND/ Am 2.Y.17/7u71 Or ScIAWC.E1 ,04

1940411CIA16 Joie kV.

(4) a ._ a ..... ..._ __

I ... I ... it. •• 1

1 . - I

.. l

- 1 ..... 1 H - a / -1

. Lv - 1

• I nwyfic Et

IV a LON VOLTAGE NiN131M6(AWASter) II r. maw vac raw renvphver ( siconDANY) E = rxrirms whvanva

(b) r . . ____ . __ are ea do■u,

I I

1 I

• 1 E i

I I I

1 1 SI

I •

I I

I 1

7

rtif a ilakAGS InfrallAteca OP IV

z*, risAnn non flora' OE il

z A- a fiance hvointecit or ar

Yir a tat / ref rovgAr ANDAwy-Tanc a

L • a a N R isammvate ce nit 7Wif1 rthvIdalif

4 1' STAR ifoolevT. T e TA An snit Art, rim*

1/6. 2 . (a) rActi vein ode A CRSCAP, TPAN.foraRitteR *NA 2 1's

sea VIVA LINCMCLE -----_

— —

1

gp.41 P4

Tana tir m3 CIP-4)

r....io irort ,

1 . 1 1 Lyn . I Tsai.* Are Pt , r

164 B. N. JAYARAM

2. THE it-UNIT CASCADE TRANSFORMER

In a ti-unit cascade transformer, n identical transformers are connected in cascade as shown in Fig. 3. The first unit is the ground unit, its tank and one end of h.v. winding being solidly grounded. The tanks of the intermediate and the last units are suitably insulated from ground for the proper voltages. Fi g. 4 shows the equivalent circuit diagram of a n-unit cascade transformer, in which each unit is represented by a three-winding transformer equivalent circuit' , and

0

teAfroosti steR

0 1 1 i

e.. -"a a I rk 71 • 4, ost .0/

A

0 - 1" ,•-• — — Tete& 1 Fir . . era

—I 1 Jr, I L I vt I .

1 121.41. sr I A 1 c , _.? 'ff

An .

i in r 1 77rtn OA S n AA

176. 3. CONNICTION DIAGRAM 0144 17- war cAscAttif MAA4,04/4 ER

is coupled to the next by an ideal 1: 1 transformer to preserve the voltage and current relations. The nth unit is represented by a two-winding transformer equivalent circuit.

The values of impedances (and admittances), currents and voltages are all those referred to thr: low voltage winding.

3. LIST OF SYMBOLS

1. V r V .A.) L2 • • • • Vu . 4 . . 11 Lo t _1 . VLn Low-side voltages of the units I to n.

2. l a = Current in the low voltage winding of the ground (No. 1) unit when the cascade set is open.

3. In- • • • 1 0-m- • • Tin-2) (n-1)/ 1 (n-1)n = Currents in the low voltage wind- ings of the units 2 to n =-- currents

in the exciting windings of the units 1 to (n 1) .

4. lx,, =---- Exciting currents of the units 1 to 1.

F

ok s

r --

s 79 a — I S t 00-40.

Q

0! I x

IM2 N / 1

1

i I k th 0

k k - - - • q - - 1-

X 1... IN --N ftwir bte ... IIII /' A' !ha 4 "till

14—. ,41 or::: . , w

1— T•i- 4

A 11

01 IN (.• to

I Ig I

w .3

it ki

1 k.d. E >s•

k •— _ _. N`-' k -..% a .k • . . a S

14 k, k

\-i%t

1.--■ if „ 0 ,115,. k kit

p ; • — ‘. k N.

I fix *

k kt) et I 4 4P 1

iP 6-9\ Ct 14 •■• /4.

0 I I P. it 1 El

b a L ..,

A ■ Li rk

-- — 0 — — at qt

A 1 CS

/..?

k 144

\ %4 it

\ 0

A +I s S3/4

s7-14 ra r • -- _ _ g• -

444 : 11"

1 11 07 I 4$ 6x A 4

W I I 0

N 6,

! L.. . g 3 >. k

c s . . • N . :::,1

• — — • r•-• • Pt 04 4; A I I t e%

a I

W I CI

I § m •

I- P- - - • ' a 1-: .

z

Vin, V02. . .V0 VH(n—Th

1 01% 1 H2. • • • 1 114. 1 1-1(n-1)•

V101. V20. • • VI(.0 V(n-1)01

5.

6.

7.

VH„ = Secondary voltages (referred).

(high voltage winding) of. the units . 1 to n

I M. = Currents in the high voltage windings of the units I to n (referred).

V„„ =--- 1-1.V. terminal to ground (referred).

voltages with respect of the units 1 to n

166

B. N. JAYARAM

10.

Yna = Tank-to-ground admittances of the units 2 to n (referred).

1,4 = Currents through the respective tank- to-ground admittances (referred).

Ige The short-circuit current flowing from the h.v. terminal of the nth unit to ground when the cascade set is shorted to ground (referred).

8. Y2G • • • YkG Y(n-1)G%

9. 1 2G • • • • I ke 1 (5-1)61

4. OPEN-CIRCUIT ADMITTANCE OF A n-UNIT CASCADE TRANSFORMER

While deriving the expression for the open-circuit admittance of a n-unit cascade transformer all series impedances are neglected and the modified equiva- lent circuit diagram is shown in Fig. 5.

Referring to the equivalent network shown in Fig. 5,

1 Open-circuit admittance =-- .„--- 12-

v Ll

and by inspection, Ifin = 0

(4.01)

(4.02)

Considering the junction S. in the nth unit,

1(18-1)11 = 'Mn + IXn Similarly at

S(n-1)) 1(i.-2) (n-1) 1.-4"-- I(n—Ups + IH(n-1) 4-

. . (4.03)

Sky 1(k-1)k

-= l k (kt-1) + 1 11k ± 'Xi: . .

= 1 •4 + 62 ± 1X2

= 1 12 4 1 H1 + 1X1

1: "A 4i ouA

i r —" — 0 9 z -- I rt r z

si I n 0

R- o ni sl

f K- 1...... i. -1 .._

_06-w A

ii -1-4-kRielli4e

14. ..1 ....

F e r i° I r 4?* 1,6014)

I;- A- sk I

117 1 i NM tb

Lir Pz:s 95‘ kb CNA i

it i r e L _ __ -4 _ _ g- ...,

"---. 'I

168

B. N. JAYARAM

Adding all the equations in (4.03)

(n-1) IO = Ek lick Ixk (4. (4)

Again considering the junction T„ in the nth unit,

1 1101-1) IHn L NG

similarly at T,"„ „_1 . .

•Tk 9 = ItG (4.05)

•

T2. = 1112 i2G

Adding all the equations in (4.05)

1111 = 120 ± I36 • • • n6 • • • +LC (4.06)

I3c• • • + IkG• • • similarly 1112 +LIG

• (4.07) IkG ••• 4- InG 1H0-1) =

•

IH(n-1) + LIG

Adding (4.06) and (4.07),

(s- 1) ft

Ek = (lc — I) I kc (4.08) 1 2

Substituting in (4.04)

= — 1 ) I ix ±k1Xk (4.09)

2 1

Further,

lJkck — 1 ) lekG Oc — 1)V0-1)0.YkG (4.10) 2

where Vor..310 = V10 VH2 + VH(ft-1)

but since V10 = V112 = = VH(k_i )

V(k-1)0 = (k 1 ) V10

.• . Ek, (k 1) IkG = VI0 — 1) 2 . YkG (4.11)

2 2

Constants of a n-Unit Cascade Transformer 169 And

T k IXk =se ill ); , Yx

1 (4.12)

but since VL1--= vL2=---" VLA 'V Ltt

St !xi.= n VIA • YX 1 (4.13)

Substituting (4.11) and (4.13) in (4.09)

' 10 = V 10 5, (k 1) 2 . YL-G Yid .(nYx) (4.14)

but since V 10 = VIA

Jo vL, knY.) + k(k —

1 )2 . Yla (4.15)

Substituting (4.15) in (4.01)

Open-circuit admittance of a n-unit cascade transformer } = [(nYx) -1-14 (k — 1) 2 YkG (4.16) -

2

5. SHORT-CIRCUIT IMPEDANCE OF A n-UNIT CASCADE TRANSFORMER

While deriving the expression for the short-circuit impedance of a n-unit cascade transformer all the shunt admittances are neglected and the modified equivalent circuit diagram is shown in Fig. 6. The high voltage terminal of the nth unit is shorted to ground and a current of j (referred) flows through the h.v. winding of the nth unit. The current distribution in the different windings of the other units is also shown in Fig. 6 and a current of n1, 4 flows through the low voltage winding of the ground unit.

Referring to the equivalent network shown in Fig. 6,

Short-circuit impedance =.. Vu nish

Considering the loop L.S.H.T 0 in the nth unit,

Vms Vida 1911 (4v + Zia

Again considering the loop H„_ 1 S„.. 1 E,i_1T„_1 in the (n 1)th unit

V idan = VH04-1) 1 Ise • Zli isk • ZE

Substituting (5.03) in (5.02)

(5 .01)

(5.02)

(5.03)

Vs„ VIr(s _i ) 1.11 • Zit IA • ZS 1.1. ZLV Isk Zg •

Ise (Zt.v 4)VH06-11 —

(5. 04)

•

s„,6)4041

rrr-wski 1 __

5b4 41 °tell)

r –a —1

(n-2) ish I

FIG. C. MOO/F/ED ESUIVALAtir CIRcUir -Zs/Rai:ZAN 0,A77-Divir caScADr TR4N49,0 A? /YEAr

CALL Sieviser P4014/rrAelvc A 5 An Nzat.rcritto E9it104rmei6 rirt Sneer— chfCc1/7" /MPASAM1/41C.E.3

Constants of a n - Unit Cascade Transformer 171 similarly VH0,....1, VH(4 -31) 2 la (Ztv ZE)

Vfik (n k)1„ (Z" ZE) VH04.1 ,

•

(5.05)

V112 VH1 — (n — 1)14 (Ztv ZE)

I. Adding (5.04) and (5.05)

\Inn VH1 — [1 + 2... +(n _ k). — 1)] 1„ (4v + ZE)

(5.06)

similarly

Viu n _10 =7= VH1 [

2... + (n (n 10 (Ztv ZE)

•

VH(I+1) = VH1 — [

(n k) ...+ (n 1)] J, (ZLV ZE)

(5.07)

•

VH2

VHI VH1

(ti 1)]I (ZLV ZE) •

4 Adding (5.06) and (5.07)

ft (us—i) EL VHk nV Hi —Ek ks (ZL, + 4) 1 1

But L.H.S. of (5.08) is zero since

fik Vak = \co =a' 0

( 5 .08)

(5.09)

(s-1.) Ek k2

I • V eb • • 1-11 n (Zis + Zs)

(5.10)

Again considering the loop L I S I II 1T1 in the first unit

Vie t = VH1 + la . Zit -4- tusk . Zjay

Dividing throughout (5.11) by n1, 6 and rearranging,

V ZH Viii ni,, ZIA/ + n • 171— .. a

.... •• -- . _

172

B. N. JA YARAM

VL , Substituting for rT; and Vffi in (5.12) from (5.01) and (5.10)

(n-1)

Li t k 2 Short-circuit impedance of a) ZIT

=-- ZIAT n + 1-n2— (Z LI, n-unit cascade transformer ZE) (5.13)

6. CONCLUSIONS

The constants of a ti-unit cascade transformer can be derived from a knowledge of the constants of any one transformer and the arrangement of the transformers. It is expected that these derived constants could be used with advantage in a two- winding transformer equivalent circuit for rapid and approximate calculation of the ratio of output voltage to input voltage of transformers connected in cascade with different load capacitances.

7. ACKNOWLEDGEMENT

The author is extremely grateful to Prof. D. J. Badkas for his keen interest and kind help in the preparation of this paper.

8. BIBLIOGRAPHY

"India's High Voltage Laboratory," Elect. Engng., N.Y., 1953, 72, 320.

"Development and characteristics of one million volt cascade transformer at California Institute of Techno- logy," J.A.I.E.E., 1925, 44, 373.

"High voltage testing equipments," 1931, 69, 898.

1. Pfestorf, G. K. M. and Badkas, D. J.

2. Sorensen, R. W.

3. Norris, E. T. and Taylor, F. W.

4.

5.

Blume, L. F. and Others

Morris, D.

.. Transformer Engineering, John Wiley and Sons, 2nd edition (1951), Chapter V—Impedance character- istics of multicircuit transformers—p. 93.

.. "Some practical equivalent circuits for multi-circuit transformers," Proc. LEE., 1951, 98 (Part II), 256.

6. Hagenguth, J. H., Rolfs, A. F. "Sixty-cycle and impulse spark-over of large gap and Degnan, W. J. spacings," Trans. A.I.E.E., 1952, 71 (Part III), 455.